[333 
137" 
[333 
:ial intersection 
¡ersecting right 
i, e, qua linear 
taken to be 
333] DEVELOPABLE FROM THE EQUATION (a, b, C, d, e$t, l) 4 = 0. 137 
or, expanding and reducing, 
A { d 2 - A + d 2 ) bd + Q x 0\ d 2 } 
+ B { e 2 — (df 2 + do 2 ) ea + d 2 d<?a?) 
+ G {!Me - {6 2 + d 2 2 ) ad - (6\ + d 2 ) be + 6A A + d 2 ) ab} 
m the quadric 
+ D { A + d 2 )ad — be— 6 X Q» ab\ = 0, 
which, if d u d 2 are the roots of the equation d 2 —£d+l = 0, and therefore 6 X + d 2 = £, 
0, 
dido = 1, and 6 2 + d 2 2 = — - 1 /, is 
A ( d 2 — £ db + d 2 ) 
3. 
+ B ( e 2 + ae + a 2 ) 
+ 0 (2de + ad — £ be + £ ab) 
+ D ( £ ad — de — ad) = 0. 
: o, 
Putting .4=6, B = 9, G = f, D = — -^, this is 
0. 
> 
9 ( a 2 + -^ ae + e 2 ) 
+ 6 ( d 2 — £ dd + d 2 ) 
+ f (£ad + 2 de + ad - £ de) 
+ J g 5 - (ad — £ ad + de) = 0, 
which is the before-mentioned quadric surface; hence the quadric surface and the cubic 
surface intersect in the two lines 
right lines, and 
(d — di d = 0, e — di 2 a = 0), (d — d 2 d = 0, e — d 2 2 a = 0) 
(where d 1} d 2 are the roots of the quadric equation d 2 — £ d x + 1 = 0); and they con 
sequently intersect also in an excubo-quartic curve, which is the theorem required to 
be proved. 
juations of any 
xirameter. But 
Blackheath, March 26, 1864. 
be written 
c. v. 
18